Introduction to Modelling and Simulation

Transcription

1 Introduction to Modelling and Simulation Prof. Cesar de Prada Dpt. Systems Engineering and Automatic Control EII, University of Valladolid, Spain

2 Digital simulation Methods and tools oriented to imitate or predict the responses of a systems against certain changes or stimulus using a computer. Modern name: digital twins

3 Modelling and Simulation Capture the behaviour of key variables of a process by means of mathematical relations and solve them in a computer with a specific purpose dx = dt y = f (x, u, p) g(x, u) Process Model Simulation 3

4 Aims of the simulation Study of a process, what if? analysis Design (process, control, ) Testing a control system before actual implementation in the plant Personnel training Operation optimization Essays in a virtual plant.. Decision support Digital Twin (real-time, simulation fed with plant data running in parallel to the real process) 4

5 Advantages of using simulation Performs changes that, if implemented in the process, will be Very costly, Too slow / fast dangerous, etc. Reproduces the experiment as many times as desired under the same conditions Saves time Provides safety Allows sensitivity studies Provides a model that can be used for many purposes Allows experimenting with systems that are not built yet 5

6 Simulation environment Training Optimization, Calibration,.. Control, design,.. Models Hardware, software Numerical methods 6

7 Models Simulation is based on mathematical models of the processes. Mathematical models are set of equations relating the variables of a process and being able to provide an adequate representation of its behaviour. They are always approximations of the real world Adequacy of a model depends on their intended use: It is a balance between accuracy in the representation of the real world and easiness in the mathematical handling of the model There are a wide variety of models according to the type of processes they represent and their aims. 7

8 Adequate representation y Proceso u time y m time Model + adequacy and facility of use in the intended application time 8

9 State space models d x(t) = f (x(t),u(t),t) dt y(t) = g(x, u(t), t) Manipulated variables and disturbances u x y Model responses x States 9

10 Continuous versus discrete event systems q h Continuous processes: Their variables evolve continuously in time and can have any value within a certain range Discrete event systems: Their variables only change values at certain time instants and they take values only from a discrete set of values 10

11 Continuous versus discrete event systems Continuous processes: Described usually by means of DAEs or PDEs. Main interest: Compute the trajectory of some variables Discrete events processes: Described usually as sequences of activities. Main interest: Statistical behavior of some variables 11

12 Discrete events simulation Oriented to describe and solve the evolution of processes described by a set of ordered activities (activity network), and provide tools for analyzing the statistical behavior of some variables Examples: Single objects Manufacturing (shoes, cars,..) Maintenance systems Offices Hospitals Services,. 12

13 Example: Telephone switchboard Calls arrive with a certain statistical distribution. If there is no line available, the call is retain waiting to be served. Which is the average waiting time? How many lines should be installed, so that the average waiting time is less than a given threshold? 13

14 Example: Manufacturing shopflow The products follow a set of stages, with a certain elapsed time in each machine, queuing to be treated until the next machine id free. Which is the average length of each queue? How many machines should be installed in parallel at each stage to keep the average queue length below a certain value? 14

15 Tuna canning factory 15

16 Time Main elements of discrete-events Sequence of tasks simulation Events that activate the tasks or are activated by the tasks Entities involved in the tasks Resources required, generated or consumed in the tasks Waiting queues, storage rooms, etc. 16

17 Main elements of discrete-events simulation Tasks: load, move, unload Entities: boxes Resources: People, trucks,.. Events: Boxes arrival, Green/Red light truck busy,... 17

18 Activity network Ordered sequence of tasks activated by time or event 18

19 Discrete-Events models Description of the set of logic relations governing the interactions among the systems elements Places Transitions Petri Nets GRAFCET Queues, state conditions, Resources Condition fulfilled 19

20 Discrete-events simulation Task network Boundary conditions Results Executed N times according to the statistical distribution given by the boundary conditions Results of the N simulations are analysed in order to infer its statistical properties 20

21 Software MicroSaint, Witness, ARENA SIMIO Animation Useful for debugging Important for presentation and explotation of the simulation Requires time and resources 21

22 Continuous processes 22

23 Static and dynamic models q q = kρ h h Static model: Gives relations among the variables corresponding to an equilibrium state Aρ d h d t = q kρ h Dynamic model: Gives relations among the variables over time 23

24 Dynamic response h q time 24

25 Static and dynamic models Static models: Describe systems in equilibrium Algebraic equations Design oriented Dynamic models in continuous time Describe the time evolution of a system ODEs, DAEs and PDEs General purpose use 25

26 Sampled data models Computer u(kt) D/A Process y(t) y(kt) A/D Discrete time models They relate the process variables at the sampling times kt Equations in differences y((k+1)t) = f( y(kt), u(kt) ) K = 0, 1, 2, 26

27 Methods to obtain models Using knowledge, nature laws, reasoning, Using experimentation and data analysis 27

28 First principles models There are obtained using knowledge on the system and reasoning based on the application of laws of nature (mass and energy balances, laws of the application, domain, ) There are based on a set of hypothesis on the system They have general validity if the hypothesis are respected Development of the models requires good knowledge of the system and good command of the physical laws. 28

29 Example: Tank Parameters q F A cross section ρ density, k constant h Mass conservation Accumulation= inflow q outflow F dm dt m dh A dt = qρ Fρ = Aρh = q k F h = k Non-linear differential equation h V = Ah Algebraic equation 29

30 Identification The model is obtained from input-output data collected from experiments, using some type of model fitting U t U Process Y Y t Model 30

31 Practical models + A practical model is always a mixture of knowledge and experimentation 31

32 Modelling / Simulation Under which conditions can the model be used? Model aims, precision, Coded for a certain computation Has been solved adequately? Hypothesis Modelling Parameterization Coding Experimentation Knowledge Experiments Simulation language All stages must be validated 32

33 Modelling methodology Knowledge of the process Modelling requires specific techniques as well as human and material resources time Aims, model type Hypothesis Model formulation Parameter estimation Model evaluation Explotation 33

34 Hypothesis c i q q c h h d Vc d t d V d t = qc i = q F Fc V F Perfect mixture = Ah c( t) dv dt F Plug flow h Ah V = ci ( t ) = ci ( t ) = ci ( t ) v Av F = q F V = Ah 34

35 Continuous model types Lump models Distributed parameters No-linear Linear Time Frequency. 35